Step unclear in Peebbles' Probability Book

[snatchingthepi]

TL;DR

There's a step from one line to another that isn't clear to me.

If anyone could help me understand how Peebles gets from line one of the autocorrelation to the second line, I'd be most grateful. I don't understand what identity or property is being used to go from a product in the expectation value to a sum in the expectation value.

 

[member 587159]

I'm not sure I understand the step you are struggling with. Is it where they calculate $R_{X,X}(t,t+\tau)$?

In that case, the formula

$$\cos(a) \cos(b)= 1/2(\cos(a+b)+\cos(a-b))$$ can be useful.

 

[snatchingthepi]

Sorry for the lateness of my thanks, but #gradschool. Yeah I feel slightly stupid now for not realizing that. It was exactly what I needed. Thank you.